What if you were given two points that a line passes through like (-1, 0) and (2, 2)? How could you find the slope of that line? After completing this Concept, you'll be able to find the slope of any line.

Watch This

Guidance

Wheelchair ramps at building entrances must have a slope between
and
. If the entrance to a new office building is 28 inches off the ground, how long does the wheelchair ramp need to be?

We come across many examples of slope in everyday life. For example, a slope is in the pitch of a roof, the grade or incline of a road, or the slant of a ladder leaning on a wall. In math, we use the word
slope
to define steepness in a particular way.

To make it easier to remember, we often word it like this:

In the picture above, the slope would be the ratio of the
height
of the hill to the horizontal
length
of the hill. In other words, it would be
, or 0.75.

If the car were driving to the
right
it would
climb
the hill - we say this is a positive slope. Any time you see the graph of a line that goes up as you move to the right, the slope is
positive
.

If the car kept driving after it reached the top of the hill, it might go down the other side. If the car is driving to the
right
and
descending
, then we would say that the slope is
negative
.

Here’s where it gets tricky: If the car turned around instead and drove back down the left side of the hill, the slope of that side would still be positive. This is because the rise would be -3, but the run would be -4 (think of the
axis - if you move from right to left you are moving in the
negative
direction). That means our slope ratio would be
, and the negatives cancel out to leave 0.75, the same slope as before. In other words, the slope of a line is the same no matter which direction you travel along it.

Find the Slope of a Line

A simple way to find a value for the slope of a line is to draw a right triangle whose hypotenuse runs along the line. Then we just need to measure the distances on the triangle that correspond to the rise (the vertical dimension) and the run (the horizontal dimension).

Example A

Find the slopes for the three graphs shown.

Solution

There are already right triangles drawn for each of the lines - in future problems you’ll do this part yourself. Note that it is easiest to make triangles whose vertices are
lattice points
(i.e. points whose coordinates are all integers).

a) The rise shown in this triangle is 4 units; the run is 2 units. The slope is
.

b) The rise shown in this triangle is 4 units, and the run is also 4 units. The slope is
.

c) The rise shown in this triangle is 2 units, and the run is 4 units. The slope is
.

Example B

Find the slope of the line that passes through the points (1, 2) and (4, 7).

Solution

We already know how to graph a line if we’re given two points: we simply plot the points and connect them with a line. Here’s the graph:

Since we already have coordinates for the vertices of our right triangle, we can quickly work out that the rise is
and the run is
(see diagram). So the slope is
.

If you look again at the calculations for the slope, you’ll notice that the 7 and 2 are the
coordinates of the two points and the 4 and 1 are the
coordinates. This suggests a pattern we can follow to get a general formula for the slope between two points
and
:

Slope between
and

or

In the second equation the letter
denotes the slope (this is a mathematical convention you’ll see often) and the Greek letter delta
means
change
. So another way to express slope is
change in
divided by
change in
. In the next section, you’ll see that it doesn’t matter which point you choose as point 1 and which you choose as point 2.

Find the Slopes of Horizontal and Vertical lines

Example C

Determine the slopes of the two lines on the graph below.

Solution

There are 2 lines on the graph:
and
.

Let’s pick 2 points on line
—say,
and
—and use our equation for slope:

If you think about it, this makes sense - if
doesn’t change as
increases then there is no slope, or rather, the slope is zero. You can see that this must be true for all horizontal lines.

Horizontal lines (
=
constant
) all have a slope of 0.

Now let’s consider line
. If we pick the points
and
, our slope equation is
. But dividing by zero isn’t allowed!

In math we often say that a term which involves division by zero is
undefined.
(Technically, the answer can also be said to be infinitely large—or infinitely small, depending on the problem.)

The slope (or
rate of change
) of a distance-time graph is a
velocity.

Guided Practice

Find the slopes of the lines on the graph below.

Solution

Look at the lines - they both slant down (or decrease) as we move from left to right. Both these lines have
negative slope.

The lines don’t pass through very many convenient lattice points, but by looking carefully you can see a few points that look to have integer coordinates. These points have been circled on the graph, and we’ll use them to determine the slope. We’ll also do our calculations twice, to show that we get the same slope whichever way we choose point 1 and point 2.

For Line
:

For Line

You can see that whichever way round you pick the points, the answers are the same. Either way,
Line
has slope -0.364, and Line
has slope -1.375.

Practice

Use the slope formula to find the slope of the line that passes through each pair of points.

(-5, 7) and (0, 0)

(-3, -5) and (3, 11)

(3, -5) and (-2, 9)

(-5, 7) and (-5, 11)

(9, 9) and (-9, -9)

(3, 5) and (-2, 7)

(2.5, 3) and (8, 3.5)

For each line in the graphs below, use the points indicated to determine the slope.

For each line in the graphs above, imagine another line with the same slope that passes through the point (1, 1), and name one more point on that line.